where \(\Delta _{yy^{\prime }}\)  is some set of margins between labels \(y\)  and \(y^{\prime }\) .
Examples of such \(\Delta \) 
include
\(\Delta _{yy^{\prime }} = \frac{1}{\mathbb {P}(y)^{1/4}}\)  [1]},
\(\Delta _{yy^{\prime }} = {\mathbb {P}(y^{\prime })}\)  [2]},
and
\(\Delta _{yy^{\prime }} = \log \frac{\mathbb {P}(y^{\prime })}{\mathbb {P}(y)}\)  [3]}, [4]}, [5]}.
Intuitively, such a loss can be seen as a soft approximation to
\(\max _{y^{\prime }\ne y} [ \Delta _{yy^{\prime }} + f_{y^{\prime }}(x)-f_{y}(x) ]_+\) , where \([a]_+\stackrel{.}{=}\max (0, a)\) ,
and thus encourages a sufficiently large gap between the logits for \(y\)  and \(y^{\prime }\) .
By ensuring that \(\Delta _{yy^{\prime }}\)  is large for rare “positive” labels \(y\)  and/or dominant “negative” labels \(y^{\prime }\) ,
one mitigates confusing a rare label for a dominant one.
In the sequel, we shall primarily be interested in the choice \(\Delta _{yy^{\prime }} = \log \frac{\mathbb {P}(y^{\prime })}{\mathbb {P}(y)}\) , which possesses good empirical performance compared to alternatives [3]}, [4]}, [5]}.
